WILD SPECTRAL BEHAVIOUR OF ANHARMONIC OSCILLATORS

E. B. DAVIES

doi:10.1112/S0024609300007050

Abstract

We show that the norms of the sequence of spectral projections of certain non-self-adjoint ordinary differential operators cannot be polynomially bounded. The proof involves the investigation of ‘tame’ bi-orthogonal systems in Banach spaces.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Davies, E.B 2001. Spectral properties of random non-self-adjoint matrices and operators. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 457, Issue. 2005, p. 191.

Davies, E.B. 2007. An Indefinite Convection-Diffusion Operator. LMS Journal of Computation and Mathematics, Vol. 10, Issue. , p. 288.

Almog, Y. 2008. The Stability of the Normal State of Superconductors in the Presence of Electric Currents. SIAM Journal on Mathematical Analysis, Vol. 40, Issue. 2, p. 824.

Ishkin, Kh. K. 2009. On the spectral instability of the Sturm-Liouville operator with a complex potential. Differential Equations, Vol. 45, Issue. 4, p. 494.

HELFFER, B. 2011. ON PSEUDO-SPECTRAL PROBLEMS RELATED TO A TIME-DEPENDENT MODEL IN SUPERCONDUCTIVITY WITH ELECTRIC CURRENT. Confluentes Mathematici, Vol. 03, Issue. 02, p. 237.

Henry, Raphaël 2012. Spectral instability of some non-selfadjoint anharmonic oscillators. Comptes Rendus Mathematique, Vol. 350, Issue. 23-24, p. 1043.

Pelloni, B. and Smith, D. A. 2013. Spectral theory of some non-selfadjoint linear differential operators. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 469, Issue. 2154, p. 20130019.

Duncan, A. and Niedermaier, M. 2013. Temporal breakdown and Borel resummation in the complex Langevin method. Annals of Physics, Vol. 329, Issue. , p. 93.

Hitrik, Michael Sjöstrand, Johannes and Viola, Joe 2013. Resolvent estimates for elliptic quadratic differential operators. Analysis & PDE, Vol. 6, Issue. 1, p. 181.

Henry, Raphaël 2014. Spectral Projections of the Complex Cubic Oscillator. Annales Henri Poincaré, Vol. 15, Issue. 10, p. 2025.

Znojil, Miloslav 2015. Non‐Selfadjoint Operators in Quantum Physics. p. 7.

Krejčiřík, D. Siegl, P. Tater, M. and Viola, J. 2015. Pseudospectra in non-Hermitian quantum mechanics. Journal of Mathematical Physics, Vol. 56, Issue. 10,

Novák, Radek 2015. On the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential. International Journal of Theoretical Physics, Vol. 54, Issue. 11, p. 4142.

Beauchard, Karine Helffer, Bernard Henry, Raphael and Robbiano, Luc 2015. Degenerate parabolic operators of Kolmogorov type with a geometric control condition. ESAIM: Control, Optimisation and Calculus of Variations, Vol. 21, Issue. 2, p. 487.

Almog, Yaniv and Helffer, Bernard 2015. On the Spectrum of Non-Selfadjoint Schrödinger Operators with Compact Resolvent. Communications in Partial Differential Equations, Vol. 40, Issue. 8, p. 1441.

Vukadinovic, J. Dedits, E. Poje, A.C. and Schäfer, T. 2015. Averaging and spectral properties for the 2D advection–diffusion equation in the semi-classical limit for vanishing diffusivity. Physica D: Nonlinear Phenomena, Vol. 310, Issue. , p. 1.

Znojil, Miloslav 2015. Quantum control and the challenge of non-Hermitian model-building. Journal of Physics: Conference Series, Vol. 624, Issue. , p. 012011.

Mityagin, Boris and Siegl, Petr 2016. Root System of Singular Perturbations of the Harmonic Oscillator Type Operators. Letters in Mathematical Physics, Vol. 106, Issue. 2, p. 147.

Znojil, Miloslav 2017. Bound states emerging from below the continuum in a solvablePT-symmetric discrete Schrödinger equation. Physical Review A, Vol. 96, Issue. 1,

Savchuk, A. M. and Shkalikov, A. A. 2017. Spectral properties of the complex airy operator on the half-line. Functional Analysis and Its Applications, Vol. 51, Issue. 1, p. 66.

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WILD SPECTRAL BEHAVIOUR OF ANHARMONIC OSCILLATORS

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